Difference between revisions of "2017 AMC 10A Problems/Problem 7"
Supermessi (talk | contribs) (Created page with "Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia heade...") |
m (→Video Solution) |
||
| (13 intermediate revisions by 11 users not shown) | |||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? | Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip? | ||
| − | (A) 30% (B) 40% (C) 50% (D) 60% (E) 70% | + | |
| + | <math>\textbf{(A)}\ 30\%\qquad\textbf{(B)}\ 40\%\qquad\textbf{(C)}\ 50\%\qquad\textbf{(D)}\ 60\%\qquad\textbf{(E)}\ 70\%</math> | ||
| + | ==Solution== | ||
| + | |||
| + | Let <math>j</math> represent how far Jerry walked, and <math>s</math> represent how far Silvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides, | ||
| + | <math>j = 2</math> | ||
| + | Since Silvia walked the diagonal, she walked the hypotenuse of a <math>45</math>, <math>45</math>, <math>90</math> triangle with leg length <math>1</math>. Thus, | ||
| + | <math>s = \sqrt{2} = 1.414...</math> | ||
| + | We can then take | ||
| + | <math>\frac{j-s}{j} \approx \frac{2 - 1.4}{2}=0.3 \implies \boxed{\textbf{(A)}\ 30\%}</math> | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/pxg7CroAt20?feature=shared&t=222 | ||
| + | |||
| + | (TheBeautyofMath) | ||
| + | |||
| + | https://youtu.be/MUxbCp-2OYQ | ||
| + | |||
| + | ~savannahsolver | ||
| + | |||
| + | ==See Also== | ||
| + | {{AMC10 box|year=2017|ab=A|num-b=6|num-a=8}} | ||
| + | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Geometry Problems]] | ||
Latest revision as of 09:38, 14 September 2024
Contents
Problem
Jerry and Silvia wanted to go from the southwest corner of a square field to the northeast corner. Jerry walked due east and then due north to reach the goal, but Silvia headed northeast and reached the goal walking in a straight line. Which of the following is closest to how much shorter Silvia's trip was, compared to Jerry's trip?
Solution
Let 
represent how far Jerry walked, and 
represent how far Silvia walked. Since the field is a square, and Jerry walked two sides of it, while Silvia walked the diagonal, we can simply define the side of the square field to be one, and find the distances they walked. Since Jerry walked two sides,
Since Silvia walked the diagonal, she walked the hypotenuse of a 
, , 
triangle with leg length 
. Thus,
We can then take
Video Solution
https://youtu.be/pxg7CroAt20?feature=shared&t=222
(TheBeautyofMath)
~savannahsolver
See Also
| 2017 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
